Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2016
Volume
Vol. 36
Number
No. 4
Description
Journal Volume
Opuscula Mathematica
Vol. 36 (2016)
Projects
Pages
Articles
Integral and fractional equations, positive solutions, and Schaefer’s fixed point theorem
(2016) Becker, Leigh C.; Burton, Theodore Allen; Purnaras, Ioannis K.
This is the continuation of four earlier studies of a scalar fractional differential equation of Riemann-Liouville type $D^qx(t) = -f(t,x(t)), \quad \lim_{t\to 0^+} t^{1-q} x(t) = x^0 \in\Re \quad (0 \lt q \lt 1), \tag {a}$ in which we first invert it as a Volterra integral equation $x(t)=x^0 t^{q-1} -\frac{1}{\Gamma (q)}\int\limits^t_0 (t-s)^{q-1}f(s,x(s))\,ds \tag {b}$ and then transform it into $\begin{multline}x(t)=x^0t^{q-1}-\int\limits^t_0 R(t-s)x^0s^{q-1}ds\\+\int\limits^t_0R(t-s) \bigg[x(s)-\frac{f(s,x(s))}{J} \bigg] ds, \tag {c}\end{multline}$ where $R$ is completely monotone with $\int^{\infty}_0 R(s)\,ds =1$ and $J$ is an arbitrary positive constant. Notice that when x is restricted to a bounded set, then by choosing J large enough, we can frequently change the sign of the integrand in going from $(b)$ to $(c)$. Moreover, the same kind of transformation will produce a similar effect in a wide variety of integral equations from applied mathematics. Because of that change in sign, we can obtain an a priori upper bound on solutions of $(b)$ with a parameter $\lambda \in (0,1]$ and then obtain an a priori lower bound on solutions of $(c)$. Using this property and Schaefer's fixed point theorem, we obtain positive solutions of an array of fractional differential equations of both Caputo and Riemann-Liouville type as well as problems from turbulence, heat transfer, and equations of logistic growth. Very simple results establishing global existence and uniqueness of solutions are also obtained in the same way.
Some stability conditions for scalar Volterra difference equations
(2016) Berezanskij, Leonid Mironovič; Migda, Małgorzata; Schmeidel, Ewa
New explicit stability results are obtained for the following scalar linear difference equation $x(n+1)-x(n)=-a(n)x(n)+\sum_{k=1}^n A(n,k)x(k)+f(n)$ and for some nonlinear Volterra difference equations.
A remark on the intersections of subanalytic leaves
(2016) Denkowski, Maciej
We discuss a new sufficient condition – weaker than the usual transversality condition – for the intersection of two subanalytic leaves to be smooth. It involves the tangent cone of the intersection and, as typically non-transversal, it is of interest in analytic geometry or dynamical systems. We also prove an identity principle for real analytic manifolds and subanalytic functions.
On the Baire classification of continuous mappings defined on products of Sorgenfrey lines
(2016) Karlova, Olena Oleksïïvna; Fodčuk, Olga Stepanívna
We study the Baire measurability of functions defined on $\mathbb{R}^T$ which are continuous with respect to the product topology on a power $\mathbb{S}^T$ of Sorgenfrey lines.
Uniform approximation by polynomials with integer coefficients
(2016) Lipnicki, Artur
Let $r$, $n$ be positive integers with $n\ge 6r$. Let $P$ be a polynomial of degree at most n on $[0,1]$ with real coefficients, such that $P^{(k)}(0)/k!$ and $P^{(k)}(1)/k!$ are integers for $k=0,\dots,r-1$. It is proved that there is a polynomial $Q$ of degree at most $n$ with integer coefficients such that $|P(x)-Q(x)|\le C_1C_2^r r^{2r+1/2}n^{-2r}$ for $x\in[0,1]$, where $C_1$, $C_2$ are some numerical constants. The result is the best possible up to the constants.